3Chapter 4 ObjectivesDetermine sample spaces and find the probability of an event, using classical probability or empirical probability.Find the probability of compound events, using the addition rules.Find the probability of compound events, using the multiplication rules.Find the conditional probability of an event.Bluman, Chapter 43

4Chapter 4 ObjectivesFind total number of outcomes in a sequence of events, using the fundamental counting rule.Find the number of ways that r objects can be selected from n objects, using the permutation rule.Find the number of ways for r objects selected from n objects without regard to order, using the combination rule.Find the probability of an event, using the counting rules.Bluman, Chapter 44

5ProbabilityProbability can be defined as the chance of an event occurring. It can be used to quantify what the “odds” are that a specific event will occur. Some examples of how probability is used everyday would be weather forecasting, “75% chance of snow” or for setting insurance rates.Bluman, Chapter 45

64-1 Sample Spaces and ProbabilityA probability experiment is a chance process that leads to well-defined results called outcomes.An outcome is the result of a single trial of a probability experiment.A sample space is the set of all possible outcomes of a probability experiment.An event consists of outcomes.Bluman, Chapter 46

15Sample Spaces and ProbabilityClassical probability uses sample spaces to determine the numerical probability that an event will happen and assumes that all outcomes in the sample space are equally likely to occur.Bluman, Chapter 415

16Sample Spaces and ProbabilityRounding Rule for ProbabilitiesProbabilities should be expressed as reduced fractions or rounded to two or three decimal places. When the probability of an event is an extremely small decimal, it is permissible to round the decimal to the first nonzero digit after the decimal point.Bluman, Chapter 416

18Example 4-6: Gender of ChildrenIf a family has three children, find the probability that two of the three children are girls.Sample Space:BBB BBG BGB BGG GBB GBG GGB GGGThree outcomes (BGG, GBG, GGB) have two girls.The probability of having two of three children being girls is 3/8.Bluman, Chapter 4

23Example 4-10: Finding ComplementsFind the complement of each event.EventComplementof the EventRolling a die and getting a 4Getting a 1, 2, 3, 5, or 6Selecting a letter of the alphabetGetting a consonant (assume y is aand getting a vowelconsonant)Selecting a month and getting aGetting February, March, April, May,month that begins with a JAugust, September, October,November, or DecemberSelecting a day of the week andGetting Saturday or Sundaygetting a weekdayBluman, Chapter 4

25Example 4-11: Residence of PeopleIf the probability that a person lives in an industrialized country of the world is , find the probability that a person does not live in an industrialized country.Bluman, Chapter 4

29Example 4-13: Blood TypesIn a sample of 50 people, 21 had type O blood, 22 had type A blood, 5 had type B blood, and 2 had type AB blood. Set up a frequency distribution and find the following probabilities.a. A person has type O blood.TypeFrequencyA22B5AB2O21Total 50Bluman, Chapter 4

30Example 4-13: Blood TypesIn a sample of 50 people, 21 had type O blood, 22 had type A blood, 5 had type B blood, and 2 had type AB blood. Set up a frequency distribution and find the following probabilities.b. A person has type A or type B blood.TypeFrequencyA22B5AB2O21Total 50Bluman, Chapter 4

31Example 4-13: Blood TypesIn a sample of 50 people, 21 had type O blood, 22 had type A blood, 5 had type B blood, and 2 had type AB blood. Set up a frequency distribution and find the following probabilities.c. A person has neither type A nor type O blood.TypeFrequencyA22B5AB2O21Total 50Bluman, Chapter 4

32Example 4-13: Blood TypesIn a sample of 50 people, 21 had type O blood, 22 had type A blood, 5 had type B blood, and 2 had type AB blood. Set up a frequency distribution and find the following probabilities.d. A person does not have type AB blood.TypeFrequencyA22B5AB2O21Total 50Bluman, Chapter 4

37Example 4-15: Rolling a DieDetermine which events are mutually exclusive and which are not, when a single die is rolled.a. Getting an odd number and getting an even numberGetting an odd number: 1, 3, or 5Getting an even number: 2, 4, or 6Mutually ExclusiveBluman, Chapter 4

38Example 4-15: Rolling a DieDetermine which events are mutually exclusive and which are not, when a single die is rolled.b. Getting a 3 and getting an odd numberGetting a 3: 3Getting an odd number: 1, 3, or 5Not Mutually ExclusiveBluman, Chapter 4

39Example 4-15: Rolling a DieDetermine which events are mutually exclusive and which are not, when a single die is rolled.c. Getting an odd number and getting a number less than 4Getting an odd number: 1, 3, or 5Getting a number less than 4: 1, 2, or 3Not Mutually ExclusiveBluman, Chapter 4

40Example 4-15: Rolling a DieDetermine which events are mutually exclusive and which are not, when a single die is rolled.d. Getting a number greater than 4 and getting a number less than 4Getting a number greater than 4: 5 or 6Getting a number less than 4: 1, 2, or 3Mutually ExclusiveBluman, Chapter 4

42Example 4-18: Political AffiliationAt a political rally, there are 20 Republicans, 13 Democrats, and 6 Independents. If a person is selected at random, find the probability that he or she is either a Democrat or an Independent.Mutually Exclusive EventsBluman, Chapter 4

44Example 4-21: Medical StaffIn a hospital unit there are 8 nurses and 5 physicians; 7 nurses and 3 physicians are females.If a staff person is selected, find the probability that the subject is a nurse or a male.StaffFemalesMalesTotalNursesPhysicians718325Total10313Bluman, Chapter 4

454.3 Multiplication RulesTwo events A and B are independent events if the fact that A occurs does not affect the probability of B occurring.Bluman, Chapter 445

47Example 4-23: Tossing a CoinA coin is flipped and a die is rolled. Find the probability of getting a head on the coin and a 4 on the die.This problem could be solved using sample space.H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6Bluman, Chapter 4

49Example 4-26: Survey on StressA Harris poll found that 46% of Americans say they suffer great stress at least once a week. If three people are selected at random, find the probability that all three will say that they suffer great stress at least once a week.Bluman, Chapter 4

51Example 4-28: University CrimeAt a university in western Pennsylvania, there were 5 burglaries reported in 2003, 16 in 2004, and 32 in If a researcher wishes to select at random two burglaries to further investigate, find the probability that both will have occurred in 2004.Bluman, Chapter 4

524.3 Conditional ProbabilityConditional probability is the probability that the second event B occurs given that the first event A has occurred.Bluman, Chapter 452

54Example 4-33: Parking TicketsThe probability that Sam parks in a no-parking zone and gets a parking ticket is 0.06, and the probability that Sam cannot find a legal parking space and has to park in the no-parking zone is On Tuesday, Sam arrives at school and has to park in a no-parking zone. Find the probability that he will get a parking ticket.N= parking in a no-parking zone, T= getting a ticketBluman, Chapter 4

56Example 4-34: Women in the MilitaryA recent survey asked 100 people if they thought women in the armed forces should be permitted to participate in combat. The results of the survey are shown.Bluman, Chapter 4

57Example 4-34: Women in the Militarya. Find the probability that the respondent answered yes (Y), given that the respondent was a female (F).Bluman, Chapter 4

58Example 4-34: Women in the Militaryb. Find the probability that the respondent was a male (M), given that the respondent answered no (N).Bluman, Chapter 4

60Example 4-37: Bow TiesThe Neckware Association of America reported that 3% of ties sold in the United States are bow ties (B). If 4 customers who purchased a tie are randomly selected, find the probability that at least 1 purchased a bow tie.Bluman, Chapter 4

614.4 Counting RulesThe fundamental counting rule is also called the multiplication of choices.In a sequence of n events in which the first one has k1 possibilities and the second event has k2 and the third has k3, and so forth, the total number of possibilities of the sequence will bek1 · k2 · k3 · · · knBluman, Chapter 461

63Example 4-39: Paint ColorsA paint manufacturer wishes to manufacture several different paints. The categories includeColor: red, blue, white, black, green, brown, yellowType: latex, oilTexture: flat, semigloss, high glossUse: outdoor, indoorHow many different kinds of paint can be made if you can select one color, one type, one texture, and one use?Bluman, Chapter 4

64Counting RulesFactorial is the product of all the positive numbers from 1 to a number.Permutation is an arrangement of objects in a specific order. Order matters.Bluman, Chapter 464

65Counting RulesCombination is a grouping of objects. Order does not matter.Bluman, Chapter 465

67Example 4-42: Business LocationsSuppose a business owner has a choice of 5 locations in which to establish her business. She decides to rank each location according to certain criteria, such as price of the store and parking facilities. How many different ways can she rank the 5 locations?Using factorials, 5! = 120.Using permutations, 5P5 = 120.Bluman, Chapter 4

68Example 4-43: Business LocationsSuppose the business owner in Example 4–42 wishes to rank only the top 3 of the 5 locations. How many different ways can she rank them?Using permutations, 5P3 = 60.Bluman, Chapter 4

70Example 4-44: Television News StoriesA television news director wishes to use 3 news stories on an evening show. One story will be the lead story, one will be the second story, and the last will be a closing story. If the director has a total of 8 stories to choose from, how many possible ways can the program be set up?Since there is a lead, second, and closing story, we know that order matters. We will use permutations.Bluman, Chapter 4

72Example 4-45: School Musical PlaysA school musical director can select 2 musical plays to present next year. One will be presented in the fall, and one will be presented in the spring. If she has 9 to pick from, how many different possibilities are there?Order matters, so we will use permutations.Bluman, Chapter 4

74Example 4-48: School MusicalsA newspaper editor has received 8 books to review. He decides that he can use 3 reviews in his newspaper. How many different ways can these 3 reviews be selected?The placement in the newspaper is not mentioned, so order does not matter. We will use combinations.Bluman, Chapter 4

76Example 4-49: Committee SelectionIn a club there are 7 women and 5 men. A committee of 3 women and 2 men is to be chosen. How many different possibilities are there?There are not separate roles listed for each committee member, so order does not matter. We will use combinations.There are 35·10=350 different possibilities.Bluman, Chapter 4

774.5 Probability and Counting RulesThe counting rules can be combined with the probability rules in this chapter to solvemany types of probability problems.By using the fundamental counting rule, the permutation rules, and the combination rule, you can compute the probability of outcomes of many experiments, such as getting a full house when 5 cards are dealt or selecting a committee of 3 women and 2 men from a club consisting of 10 women and 10 men.Bluman, Chapter 477

79Example 4-52: Committee SelectionA store has 6 TV Graphic magazines and 8 Newstime magazines on the counter. If two customers purchased a magazine, find the probability that one of each magazine was purchased.TV Graphic: One magazine of the 6 magazinesNewstime: One magazine of the 8 magazinesTotal: Two magazines of the 14 magazinesBluman, Chapter 4

81Example 4-53: Combination LocksA combination lock consists of the 26 letters of the alphabet. If a 3-letter combination is needed, find the probability that the combination will consist of the letters ABC in that order. The same letter can be used more than once. (Note: A combination lock is really apermutation lock.)There are 26·26·26 = 17,576 possible combinations.The letters ABC in order create one combination.Bluman, Chapter 4